Lecture Notes in Physics

Transcription

1 Heidelberg New.York Lecture Notes in Physics Edited by J. Ehlers, Mtinchen, K. Hepp, Ztirich and H. A. Weidenmiiller, Heidelberg Managing Editor: W. Beiglback, Heidelberg 13 Michael Ryan University of North Carolina, Chapel Hill and University of Maryland, College Park Hamiltonian Cosmology Springer-Verlag Berlin l l 1972

2 ISBN Springer-Verlag Berlin. Heidelberg. New York ISBN O Springer-Vet&g New York. Heidelberg. Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 3 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the by Springer-Verlag Berlin Heidelberg Library of Congress Catalog Card Number Printed in Germany Offsetdruck: Julius Beltz, HemsbacNBergstr.

3 CONTENTS I. II. III. IV. V. VI. VII. Introduction The ADM Formalism Applied to Homogeneous Cosmologies The Hamiltonian Formalism: Simple Examples, their Classical and Quantum Behavior The Hamiltonian Formulation Applied to more Complex Systems *..*.* *..~.*...**.*...~**.* Applications to Bianchi-Type Universes..... Superspace *... Quantization Appendix A Lagrangian Cosmology Appendix B The Quantum Behavior of an Expanding, One-Dimensional, Square Well Appendix C Miscellaneous Hamiltonian Calculations 159 References

4 PREFACE Since the initial application of Hamiltonian techniques to cosmology a few years ago, the field has grown so rapidly that a review of the work that has been done, along with an explanation of work in progress, and an indication of possible future direction, has become desirable. It is hoped that this report will serve this purpose. The rapid publication of the LECTURE NOTES IN PHYSICS should insure that the information will reach the hands of interested people before the rapid development of the field makes it obselete. In a work of this sort it seems almost superfluous to thank those who have made it possible; one's indebtedness stands out on every page. I would, however, like to take special note of the help, through discussions, of Prof. C. Misner, Prof. B. DeWitt, Dr. H. Zapolsky, Dr. K. Jacobs, Dr. Y. Nutku, Messrs. D. Chitre, L. Fishbone, and V. Moncrief. I wish to thank all of these people, as well as Mr. L. Hughston, Miss B. Kobre, and Dr. K. Kucha~ for allowing me to see prepublication drafts of their work. This report was begun at the University of Maryland and approximately half of it was finished there. The second half was completed after a move to the University of North Carolina. The format of LECTURE NOTES IN PHYSICS makes the typing and illustration of manuscripts vital. Because of this I want to express my sincere gratitude to Mrs. B. Alexander and Mrs. J. Alexander of the University of Maryland, and Mrs. J. McCloud of the University of North Carolina for their patience and the beautiful typing job they have done. I would also like to thank Mrs. J. Hudson for a set of excellent figures.

5 IV I am grateful for financial support from NASA grant NGR at the University of Maryland and an NSF grant to Prof. B.S. DeWitt at the University of North Carolina.

6 MATHEMATICAL FOREWORD While conventional symbols are used throughout for most quantities (eg. RV~aB for the Riemann tensor), most of the mathematical operations are carried out in non-coordinate frames and much of the notation is that of modern differential geometry. We give below a list of references for those unfamiliar with techniques and notation. There is a possibility of confusion between some of the notation of the calculus and that of modern differential geometry (for instance, d is used in its usual sense (dx/dt) and also to represent the operator of exterior differentiation), but the meaning of symbols should be clear from context if one is sufficiently familiar with differential geometry. Two special points of notation need to be noted. We shall dis- tinguish differential one-forms (but not n-forms for n>~ by writing them with a tilde and vectors (in both the modern differential geometry sense and the usual sense) by superior arrows (i.e. ~). Second, we shall use the notation ~/dt for a matrix A to mean i da SA d-~ = ~ [A -1 ~-~ + ~ A -1] (& = g {A-l (da) + (da)a-1}). References for Modern Differential Geometry: See C. Misner in Astrophysics and General R~lativity (Vol. I), edited by M. Chretien, S. Deser and J. Goldstein (Gordon and Breach, New York, 1969) for the immediate antecedents to the ideas and notation used in the present work.

7 VI The classic work is: E. Cartan, Lemons sur la geometrie des espaces de Riemann, (Gauthier-Villars, Paris, 1951). Other valuable works are: H. Flanders, Differential Forms, (Academic Press, New York, 1963), N. Hicks, Notes on Differential Geometry, (Van Nostrand Mathematical Studies #3), (D. Van Nostrand, Princeton, N. J., 1965), T. Willmore, Introduction to Differential Geometry, (Oxford University Press, Oxford, 1959).